Deform a hole on a torus
This can be done by building the torus from 8 different vertex objects, each with 2 Screw modifiers.
One Screw modifier for making a circular section of the torus, and another one to rotate or loop this circle around a center (with an offset) thus creating a torus. The torus turning into 2 rings, and the 4 corners of the hole, warrants the total number of vertex objects and modifiers.

Linking the Screw modifier Angle values with drivers allows a single Angle value to deform the torus.
Making the inverse of this was my starting point so all the drivers are linked to a modifier on the initial vertex I used to make this. In hindsight a bit of a weird place but it works just the same. (So changing the Angle value of the top modifier on the &quot;Vert&quot; object makes it all move.)
I intent to elaborate on this solution a bit when i have more time (especially if you are new to Blender, drivers might take some explaining), but in the meantime...
...Here is the file:

And here is another file utilizing Mirror modifiers on only (and reducing the number to) 2 vertex objects: